Check Answer: Show that two Bernoulli random variables are independent

I've found a good answer to a particular question I was given, though I'd like a second opinion on whether or not my answer is solid (I might be "begging the question" in this).

Show that two Bernoulli random variables and are independent if and only if .

Here's my answer.

Two Bernoulli random variables are independent if and only if they are uncorrelated, and thus have a covariance of zero ( ).

Let be the pmf of , and let be the pmf of .

If and are independent then, by definition,

,

as .

If on the other hand we have that , then

.

Therefore, and are independent.

Are there any mess-ups I might have made somewhere? I got this answer from this pdf file.